Metropolis walking

To evaluate an expectation value with the VMC method, a Metropolis walk is generally employed [20, 21]. The procedure begins with an initial distribution of points generated using a wave function obtained from an independent electronic structure method, followed by selection of subsequent sets of points until the collection of points is distributed as mod squared of the trial wave function, i.e.,... 

The step sizes for a typical Metropolis walk are usually chosen to give an acceptance ratio of about one-half in order to maximize the rate of diffusion and improve the sampling speed. Serial correlation of points is usually high. In many-dimensional (or many-electron) systems, the steps may be taken one dimension (or one electron) at a time or all at once. The optimum step sizes and/or combinations of steps depend strongly on the nature of the system treated. 

Another alternative, likely to be more efficient than Metropolis sampling, is the use of probability density functions P. These relatively simple functions, which approximate and mimic the density of the more complex function V / , can be sampled directly without a Metropolis walk and the associated serial correlation. Sample points of unit weight are obtained with probabilities proportional to the probability density P, and their weights are multiplied by the factor if /P to give overall v /o weighting. The expectation value of the energy ) is then given by... 

The atom-exchange method was developed by Tsai, Abraham, and Pound to speed barrier crossing in binary (two types of atoms) alloy cluster simulations. During the Metropolis walk two different types of atoms are periodically chosen, and their positions are exchanged. The exchange is accepted or rejected by the standard Metropolis acceptance probability. The utility of this method is naturally limited to systems of this particular type, namely, binary atomic clusters and liquids. 

In Monte Carlo minimization, an additional step is inserted after the random walk before Metropolis evaluation. This step is a... 

The crucial point to note is that the above average combines information about both the accepted and the rejected state of a trial move. Note that the Monte Carlo algorithm used to generate the random walk among the states n need not be the same as the one corresponding to TTnm- For instance, we could use standard Metropolis to generate the random walk, and use the S3Tnmetric rule [4]... 

It should be stressed that the the random walk in state space need not be generated using this rule - in fact, for the random walk, the original Metropolis rule is usually better suited. The difference in the variance of the energy of WRMC and standard MC is ... 

---